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Journal of Group Theory

Editors-in-Chief: Benjamin Klopsch and Christopher W. Parker
Managing Editors: John S. Wilson and Evgeny I. Khukhro

Language: English

First published: January 1, 1998

Publication Frequency: 6 issues per year

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The Journal of Group Theory is devoted to the publication of original research articles in all aspects of group theory. Articles concerning applications of group theory and articles from research areas which have a significant impact on group theory will also be considered.

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Group Theory
Representation Theory of Groups
Computational Aspects of Group Theory
Combinatorics and Graph Theory
Algebra and Number Theory

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Groups acting acylindrically on trees

September 23, 2025

William Daniel Cohen

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We develop a notion of groups that act acylindrically and non-elementarily on simplicial trees, which we call acylindrically arboreal groups. We then prove a complete classification of when graph products of groups and the fundamental groups of certain hyperbolic 3-manifolds are acylindrically arboreal, and use these classifications to provide examples of acylindrically hyperbolic groups that have actions on trees but have no non-elementary acylindrical actions on trees.
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The excess zero graph of a Coxeter group

August 15, 2025

Sarah Hart, Veronica Kelsey, Peter Rowley

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For a Coxeter group 𝑊 with length function ℓ, the excess zero graph E 0 ⁢ ( W ) \mathcal{E}_{0}(W) has vertex set the non-identity involutions of 𝑊, with two involutions 𝑥 and 𝑦 adjacent whenever ℓ ⁢ ( x ⁢ y ) = ℓ ⁢ ( x ) + ℓ ⁢ ( y ) \ell(xy)=\ell(x)+\ell(y) . Properties of this graph such as connectivity, diameter and valencies of certain vertices of E 0 ⁢ ( W ) \mathcal{E}_{0}(W) are explored.
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Dynamical Tits alternative for groups of almost automorphisms of trees

August 6, 2025

Martín Gilabert Vio

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We prove a dynamical variant of the Tits alternative for the group of almost automorphisms of a locally finite tree 𝒯: a group of almost automorphisms of 𝒯 either contains a nonabelian free group playing ping-pong on the boundary ∂ T \partial\mathcal{T} , or the action of the group on ∂ T \partial\mathcal{T} preserves a probability measure. This generalizes to all groups of tree almost automorphisms a result of S. Hurtado and E. Militon for Thompson’s group 𝑉, with a hopefully simpler proof.
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Finite semiprimitive permutation groups of rank 3

August 5, 2025

Cai Heng Li, Hanyue Yi, Yan Zhou Zhu

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A transitive permutation group is called semiprimitive if each of its normal subgroups is either semiregular or transitive. The class of semiprimitive groups properly includes primitive groups, quasiprimitive groups and innately transitive groups. The latter three classes of rank 3 permutation groups have been classified, making significant progress towards solving the long-standing problem of classifying permutation groups of rank 3. In this paper, we complete the classification of finite semiprimitive groups of rank 3, building on the recent work of Huang, Li and Zhu. Examples include Schur coverings of certain almost simple 2-transitive groups and three exceptional small groups.
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On the commutator subgroup of the unit group of a modular group algebra

August 5, 2025

Tibor Juhász, Meena Sahai

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Let 𝐹 be a field of odd characteristic and 𝐺 a group for which the group algebra F ⁢ G FG is modular. In this paper, necessary and sufficient conditions are presented for the commutator subgroup of the unit group of F ⁢ G FG to be nilpotent of class at most 2.
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Finite groups with 𝐻_𝜎-subnormally embedded 𝔑_𝜎-critical subgroups

August 5, 2025

Zhenfeng Wu, Chi Zhang

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Let 𝐺 be a finite group and σ = { σ i ∣ i ∈ I } \sigma=\{\sigma_{i}\mid i\in I\} a partition of the set of all primes. The group 𝐺 is said to be 𝜎-primary if 𝐺 is a σ i \sigma_{i} -group for some i ∈ I i\in I , 𝜎-nilpotent if 𝐺 is a direct product of 𝜎-primary groups, N σ \mathfrak{N}_{\sigma} -critical if 𝐺 is not 𝜎-nilpotent but all proper subgroups of 𝐺 are 𝜎-nilpotent. A subgroup 𝐻 of 𝐺 is called a 𝜎-Hall subgroup of 𝐺 provided | H | \lvert H\rvert and | G : H | \lvert G:H\rvert are 𝜎-coprime and 𝜎-subnormal in 𝐺 if there is a subgroup chain H = H 0 ≤ H 1 ≤ ⋯ ≤ H t = G H=H_{0}\leq H_{1}\leq\cdots\leq H_{t}=G such that either H i − 1 H_{i-1} is normal in H i H_{i} or H i / ( H i − 1 ) H i H_{i}/(H_{i-1})_{H_{i}} is 𝜎-primary for all i = 1 , … , t i=1,\ldots,t . A subgroup 𝐻 of 𝐺 is called H σ H_{\sigma} -subnormally embedded in 𝐺 if 𝐻 is a 𝜎-Hall subgroup of some 𝜎-subnormal subgroup of 𝐺. In this paper, we prove that the second derived subgroup of a finite group 𝐺 is 𝜎-nilpotent if every N σ \mathfrak{N}_{\sigma} -critical subgroup of 𝐺 is H σ H_{\sigma} -subnormally embedded in 𝐺.
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Infinite systems of equations in abelian and nilpotent groups

August 5, 2025

Mikhail A. Mikheenko

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Every abelian (and even every nilpotent) group contains a solution of any finite unimodular system of equations over itself. However, this is not true for infinite systems. We deduce a criterion for a periodic abelian group to contain a solution of any infinite unimodular system of equations over itself. Using this criterion, we show that nilpotent groups of bounded period also contain solutions of all infinite unimodular systems of equations over themselves. Solvability of every nonsingular infinite system of equations in each divisible nilpotent group is shown as well.
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Set-stabilizers in solvable permutation groups

July 1, 2025

David Gluck

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Let 𝐺 be a finite solvable permutation group. Then, modulo a possibly trivial normal elementary abelian 3-subgroup, some set-stabilizer in 𝐺 is a 2-group.
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Structure of an exotic 2-local subgroup in 𝐸₇(𝑞)

May 27, 2025

Mikko Korhonen

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Let 𝐺 be the finite simple group of Lie type G = E 7 ⁢ ( q ) G=E_{7}(q) , where 𝑞 is an odd prime power. Then 𝐺 is an index 2 subgroup of the adjoint group G ad G_{\operatorname{ad}} , which is also denoted by G ad = Inndiag ⁡ ( G ) G_{\operatorname{ad}}=\operatorname{Inndiag}(G) and known as the group of inner-diagonal automorphisms. It was proven by Cohen–Liebeck–Saxl–Seitz (1992) that there is an elementary abelian 2-subgroup 𝐸 of order 4 in G ad G_{\operatorname{ad}} such that N G ad ⁢ ( E ) / C G ad ⁢ ( E ) ≅ Sym 3 and C G ad ⁢ ( E ) = E × Inndiag ⁡ ( D 4 ⁢ ( q ) ) . N_{G_{\operatorname{ad}}}(E)/C_{G_{\operatorname{ad}}}(E)\cong\mathrm{Sym}_{3}\quad\text{and}\quad C_{G_{\operatorname{ad}}}(E)=E\times\operatorname{Inndiag}(D_{4}(q)). Furthermore, such an 𝐸 is unique up to conjugacy in G ad G_{\operatorname{ad}} . It is known that N G ⁢ ( E ) N_{G}(E) is always a maximal subgroup of 𝐺, and N G ad ⁢ ( E ) N_{G_{\operatorname{ad}}}(E) is a maximal subgroup of G ad G_{\operatorname{ad}} unless N G ad ⁢ ( E ) ≤ G N_{G_{\operatorname{ad}}}(E)\leq G . In this note, we describe the structure of N G ⁢ ( E ) N_{G}(E) . It turns out that we have N G ⁢ ( E ) = N G ad ⁢ ( E ) N_{G}(E)=N_{G_{\operatorname{ad}}}(E) if and only if q ≡ ± 1 mod 8 q\equiv\pm 1\bmod 8 .
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The proper geometric dimension of the mapping class group of an orientable surface with punctures

May 27, 2025

Nestor Colin, Rita Jiménez Rolland, Porfirio L. León Álvarez, Luis Jorge Sánchez Saldaña

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We show that the full mapping class group of any orientable closed surface with punctures admits a cocompact classifying space for proper actions of dimension equal to its virtual cohomological dimension. This was proved for closed orientable surfaces and for pure mapping class groups by Aramayona and Martínez Pérez. As a consequence of our result, we also obtain the proper geometric dimension of full spherical braid groups.
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A note on complex hyperbolic lattices and strict hyperbolization

May 23, 2025

Kejia Zhu

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We study the connection between the fundamental groups of complex hyperbolic manifolds and those of spaces arising from the (relative) strict hyperbolization process due to Charney–Davis and Davis–Januszkiewicz–Weinberger. Viewing a non-uniform lattice Γ in PU ⁢ ( n , 1 ) \text{PU}(n,1) as a relatively hyperbolic group with respect to its cusp subgroups in the usual way, we show that, when n ≥ 2 n\geq 2 , Γ is not isomorphic to any relatively hyperbolic group arising from the relative strict hyperbolization process, via work of Lafont–Ruffoni. We also prove that a uniform lattice in PU ⁢ ( n , 1 ) \text{PU}(n,1) is not the fundamental group of a Charney–Davis strict hyperbolization when n ≥ 2 n\geq 2 , assuming the initial complex satisfies some mild conditions.
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Strong indecomposability of the outer automorphism groups of nonabelian free profinite groups

May 22, 2025

Arata Minamide, Shota Tsujimura

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In this paper, we prove that the outer automorphism groups of nonabelian topologically finitely generated free profinite groups are strongly indecomposable. This means that no open subgroup has a nontrivial direct product decomposition.
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𝜏-Tilting finiteness of group algebras of semidirect products of abelian 𝑝-groups and abelian 𝑝′-groups

May 22, 2025

Naoya Hiramae, Yuta Kozakai

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Demonet, Iyama and Jasso introduced a new class of finite-dimensional algebras, 𝜏-tilting finite algebras . It was shown by Eisele, Janssens and Raedschelders that tame blocks of group algebras of finite groups are always 𝜏-tilting finite. Given the classical result that the representation type (representation finite, tame or wild) of blocks is determined by their defect groups, it is natural to ask what kinds of subgroups control 𝜏-tilting finiteness of group algebras or their blocks. In this paper, as a positive answer to this question, we demonstrate that 𝜏-tilting finiteness of a group algebra of a finite group 𝐺 is controlled by a 𝑝-hyperfocal subgroup of 𝐺 under some assumptions on 𝐺. We consider a group algebra of a finite group P ⋊ H P\rtimes H over an algebraically closed field of positive characteristic 𝑝, where 𝑃 is an abelian 𝑝-group and 𝐻 is an abelian p ′ p^{\prime} -group acting on 𝑃, and show that 𝑝-hyperfocal subgroups determine 𝜏-tilting finiteness of the group algebras in this case.
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On the Grossman representations of the automorphism groups of free groups

May 22, 2025

Takao Satoh

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The goal of this paper is to show that, for any k ≥ 2 k\geq 2 , the subgroup of the automorphism group of a free group consisting of all automorphisms that act trivially on the ( k + 1 ) (k+1) -th derived quotient of the free group is not finitely generated in some stable range. In order to show this, we consider a slight generalization of results of Grossman in [Representations of the automorphism groups of free groups, J. Algebra 30 (1974), 388–399], and a sequence of descending filtrations of the IA-automorphism group of a free group.
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Minimal degrees for faithful permutation representations of groups of order 𝑝⁵ where 𝑝 is an odd prime

May 22, 2025

Sunil Kumar Prajapati, Ayush Udeep

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In a 2018 paper, Behravesh and Delfani computed the minimal degree of a faithful permutation representation for groups of order p 5 p^{5} , where 𝑝 is an odd prime. However, their results are incorrect and incomplete: some groups are not considered, some minimal degrees are incorrectly determined, and the reported list of degrees is incomplete. We determine the minimal degree of a faithful permutation representation for each group of order p 5 p^{5} , where 𝑝 is an odd prime. We also record how to obtain such a representation.
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On the 𝑝-length and 𝔘-class of a 𝑝-solvable finite group

May 22, 2025

Lü Gong, Yan Yan, Baojun Li

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In finite group theory, 𝑝-length is an important invariant for a 𝑝-solvable group. The famous Hall–Higman 𝑝-length theorem states that the 𝑝-length of a 𝑝-solvable group is bounded by the nilpotent class of its Sylow 𝑝-subgroups. In this paper, we improve this result by giving a better estimate on the 𝑝-length of a 𝑝-solvable group in terms of the 𝔘-class of its Sylow 𝑝-subgroups.
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Regular saturated formations of finite soluble groups

April 25, 2025

Viachaslau I. Murashka

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For a formation 𝔉 of finite groups, consider the graph whose vertices are elements of a finite group, with two vertices connected by an edge if and only if they generate a non-𝔉-group as elements of the group. A hereditary formation 𝔉 is said to be regular if the set of all isolated vertices of the described graph coincides with the intersection of all 𝔉-maximal subgroups in every group. In this paper, a constructive description of saturated regular formations of soluble groups is given, which improves the results of Lucchini and Nemmi. In particular, it is shown that saturated regular non-empty formations of soluble groups are just hereditary formations 𝔉 of soluble groups that contain every group whose cyclic primary subgroups are all 𝐾-𝔉-subnormal. Also, we prove that the lattice of saturated regular formations of soluble groups is lattice isomorphic to the Steinitz lattice.
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Finite 2-groups with exactly three automorphism orbits

April 3, 2025

Alexander Bors, Stephen P. Glasby

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We give a complete classification of the finite 2-groups 𝐺 for which the automorphism group Aut ⁡ ( G ) \operatorname{Aut}(G) acting naturally on 𝐺 has three orbits. There are two infinite families and one additional group, of order 2 9 2^{9} . All of them are Suzuki 2-groups, and they appear (in a different context) in an earlier classification of Dornhoff.
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Finite class 2 nilpotent and Heisenberg groups

April 3, 2025

Dávid R. Szabó

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We present a structural description of finite nilpotent groups of class at most 2 using a specified number of subdirect and central products of such groups that are 2-generated. As a corollary, we show that any such group is isomorphic to a subgroup of a Heisenberg group satisfying certain properties. The motivation for these results is of a topological nature as they can be used to give lower bounds to the nilpotently Jordan property of the birational automorphism group of varieties and the homeomorphism group of compact manifolds.
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Bounding the degree of generic sharp transitivity

March 4, 2025

Tuna Altınel, Joshua Wiscons

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We show that a generically sharply 𝑡-transitive permutation group of finite Morley rank on a set of rank 𝑟 satisfies t ≤ r + 2 t\leq r+2 provided the pointwise stabilizer of a generic ( t − 1 ) (t-1) -tuple is an 𝐿-group, which holds, for example, when this stabilizer is solvable or when r ≤ 5 r\leq 5 . This makes progress towards establishing the natural bound on 𝑡 implied by the Borovik–Cherlin conjecture that every generically ( r + 2 ) (r+2) -transitive permutation group of finite Morley rank on a set of rank 𝑟 is of the form PGL r + 1 ⁡ ( F ) \operatorname{PGL}_{r+1}(F) acting naturally on P r ⁢ ( F ) \mathbb{P}^{r}(F) . Our proof is assembled from three key ingredients that are independent of the main theorem – these address actions of Alt ⁡ ( n ) \operatorname{Alt}(n) on 𝐿-groups of finite Morley rank, generically 2-transitive actions with abelian point stabilizers, and simple groups of rank 6.
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Finite imprimitive rank 3 affine groups – I

January 1, 2000

Cai Heng Li, Luyi Liu, Hanyue Yi, Yan Zhou Zhu

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This is one of a series of papers which aims to classify imprimitive affine groups of rank 3. In this paper, a complete classification is given of such groups of characteristic 𝑝 whose point stabilizers are not 𝑝-local, which shows that such groups are very rare; specifically, the only examples are the two non-isomorphic groups of the form 2 4 : GL 3 ⁢ ( 2 ) 2^{4}{:}\mathrm{GL}_{3}(2) with a unique minimal normal subgroup.
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Intersections of Sylow 2-subgroups in symmetric groups

January 1, 2000

Sean Eberhard

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We compute the asymptotic probability that a random pair of Sylow 2-subgroups in S n S_{n} or A n A_{n} intersects trivially. This calculation complements recent work of Diaconis, Giannelli, Guralnick, Law, Navarro, Sambale, and Spink.
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Certain residual properties of HNN-extensions with normal associated subgroups

January 1, 2000

Evgeny Victorovich Sokolov, Elena Alexandrovna Tumanova

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Let 𝔼 be the HNN-extension of a group 𝐵 with subgroups 𝐻 and 𝐾 associated by an isomorphism φ : H → K \varphi\colon H\to K . Suppose that 𝐻 and 𝐾 are normal in 𝐵 and ( H ∩ K ) ⁢ φ = H ∩ K (H\cap K)\varphi=H\cap K . Under these assumptions, we prove necessary and sufficient conditions for 𝔼 to be residually a 𝒞-group, where 𝒞 is a class of groups closed under taking subgroups, quotient groups, and unrestricted wreath products. Among other things, these conditions give new facts on the residual finiteness and the residual 𝑝-finiteness of the group 𝔼.
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Finite groups with few subgroups not in the Chermak–Delgado lattice

January 1, 2000

Jiakuan Lu, Xi Huang, Qinwei Lian, Wei Meng

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For a finite group 𝐺, we denote by C ⁢ D ⁢ ( G ) \mathcal{CD}(G) the Chermak–Delgado lattice of 𝐺, and by v ⁢ ( G ) v(G) the number of conjugacy classes of subgroups of 𝐺 not in C ⁢ D ⁢ ( G ) \mathcal{CD}(G) . In this paper, we determine the finite groups 𝐺 for which v ⁢ ( G ) = 1 , 2 , 3 v(G)=1,2,3 .

Journal Impact Factor	0.5	2024, Journal Citation Reports (Clarivate, 2025)
5-year Journal Impact Factor	0.5	2024, Journal Citation Reports (Clarivate, 2025)
Journal Citation Indicator	0.40	2024, Journal Citation Reports (Clarivate, 2025)
CiteScore	1.0	2024, Scopus (Elsevier B.V., 2025)
SCImago Journal Rank	0.486	2024, SJR (Scimago Lab, 2025; Data Source: Scopus)
Source Normalized Impact per Paper	0.956	2024, CWTS Journal Indicators (CWTS B.V., 2025; Data Source: Scopus)
Mathematical Citation Quotient	0.54

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References. References should be collected at the end of the paper and numbered in alphabetical order of the authors’ names. Titles of journals should be abbreviated as in Mathematical Reviews. The following examples show the preferred style for references to a book, a preprint that has appeared on arXiv, a paper in a journal, a paper
in a proceedings volume, and an unpublished dissertation.

[1] J. L. Alperin. Local representation theory (Cambridge University Press, 1986).
[2] C. Bonnafe´ and M. J. Dyer. Semidirect product decomposition of Coxeter groups. Preprint (2008). http://arxiv.org/abs/0805.4100.
[3] W. Feit and J. G. Thompson. Solvability of groups of odd order. Pacific J. Math. 13 (1963), 774–1029.
[4] G. Higman. Some non-simplicity criteria for finite groups. In Proceedings of the Second International Conference on the Theory of Groups, Lecture Notes in Math. 372 (Springer-Verlag, 1974), pp. 367–376.
[5] J. D. King. Finite presentability of Lie algebras and pro-p groups. Ph.D. thesis. University of Cambridge (1995).

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Editors-in-Chief

Benjamin Klopsch, Heinrich-Heine-Universität (Germany)
Christopher Parker, University of Birmingham (UK)

Managing Editors

Evgeny Khukhro, University of Lincoln (UK) and Sobolev Institute of Mathematics (Russia)
John S. Wilson, University of Oxford (UK)

Editorial Board

Olivier Dudas, Institut de Mathématiques de Marseille (France)
Bettina Eick, TU Braunschweig (Germany)
Michael Giudici, The University of Western Australia (Australia)
Robert Guralnick, University of Southern California (USA)
Adrian Ioana, University of California, San Diego (USA)
Benjamin Klopsch, Heinrich-Heine-Universität (Germany)
Anton Klyachko, Moscow State University (Russia)
Dessislava H. Kochloukova, Cidade Universitária "Zeferino Vaz" (Brazil)
Andrea Lucchini, Università di Padova (Italy)
Frank Lübeck, RWTH Aachen (Germany)
Cheryl Praeger, The University of Western Australia (Australia)
Rachel Skipper, University of Utah (USA)
Britta Späth, Bergische Universität Wuppertal (Germany)
Hung Tong-Viet, Binghamton University (USA)
Gareth Tracey, University of Warwick (UK)
George Willis, University of Newcastle (Australia)

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eISSN:

1435-4446

ISSN:

1433-5883

Language:

English

Publisher:

De Gruyter

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First published:

January 1, 1998

Publication Frequency:

6 issues per year

eISSN: 1435-4446

ISSN: 1433-5883

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Researchers interested in Group Theory

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